Complex Variables Q&A: Solving Quadratics and Finding Roots

[kingwinner]

Homework Statement

I'm beginning my studies in complex variables and have some questions...

Q1) We know that x²=9 => x=+/- √9 = +/- 3.
Suppose z^2 = w where z and w are COMPLEX numbers, then is it still true to say that
z = +/- √w ? Why or why not?

Q2) "Let az² + bz + c =0, where a,b,c are COMPLEX numbers, a≠0.
Then the usual quadratic formula still holds."

My concern is with the √(b²-4ac) part. How can we find √(b²-4ac) when b²-4ac is a COMPLEX number?
For example, what does √(-1+4i) mean on its own and how can we find it? I know there is a general procedure(using polar form and angles) to solve for the nth root of a complex number (z^n=w), but I still don't understand what √(-1+4i) means on its own.
Even for real numbers, there is a difference between solving x²=9 and finding √9, right? So is there any difference between finding √(-1+4i) and solving z²=-1+4i for z using polar form and angles?

Homework Equations

Complex variables

The Attempt at a Solution

As shown above.

I hope someone can explain these. Any help is much appreciated!

 

[Office_Shredder]

z = +/- √w loses meaning because there is no distinguished choice for √w. Normally we define √x to be the positive square root of x, but there's no such thing as positive and negative in the complex numbers. However what we CAN say is that if a²=w and b²=w, then a²-b²=0, and factoring (a-b)(a+b)=0. So a=b or a=-b

√(-1+4i) is just going to mean a complex number which, when squared, gives -1+4i. It doesn't specify which one, but for the quadratic formula it doesn't matter which one you pick, because they both work. When writing \pm in the quadratic formula for real numbers, what you really mean is that you can use either square root of the number and you will get a root. The same holds for complex numbers as well

 

[kingwinner]

Q2) z= [-b +/- sqrt(b^2 - 4ac)] / 2a

Say b^2 - 4ac turns out to be -1+4i
Let w^2 = -1+4i. Use polar form and angles to solve for w, we get 2 solutions, call them w1 and w2.

And you mean we can pick either one of w1, w2 as √(-1+4i), right?
Say if I pick w1, then the solutions will be
z=[-b +/- w1] / 2a

Alternatively, if I pick w2, then the solutions will be
z=[-b +/- w2] / 2a which will be the SAME as the above, right? Why are they the same?

Thanks for explaining!

 

[ehild]

Q2) z= [-b +/- sqrt(b^2 - 4ac)] / 2a

Say b^2 - 4ac turns out to be -1+4i
Let w^2 = -1+4i. Use polar form and angles to solve for w, we get 2 solutions, call them w1 and w2.

How are w1 and w2 related? Compare their polar forms.

ehild

 

[kingwinner]

I think w1 = -w2, but is this always true? Why or why not?

 

[ehild]

w₁ and w₂ can be written also in polar form:

as w=|w₁|=|w₂|,

and -1=e^iΠ

w₁=w e^îφ and w₂=-w₁=w e^i(φ+Π).

Use w₁ and w₂ in the quadratic form instead of ±.
ehild